TBH I don't think you will find many 1930's-50's chronographs of 38mm or larger very easily. Just not many made and especially not at the budget you have.
Most are 35-36mm IME and usually chrome plated cases.
At the cheaper end most will have Landeron's or some sort of Venus movement, some Venus' can be hard to get spares for, so I would recommend getting something in good working order.
Probably best to stay away from Pierce unless working fine too.
I actually like chrono's from that evocative period and have a few.